What is Fourier transform: Definition + 1000 Threads
In mathematics, a Fourier transform (FT) is a mathematical transform that decomposes functions depending on space or time into functions depending on spatial or temporal frequency, such as the expression of a musical chord in terms of the volumes and frequencies of its constituent notes. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time.
The Fourier transform of a function of time is a complex-valued function of frequency, whose magnitude (absolute value) represents the amount of that frequency present in the original function, and whose argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is not limited to functions of time, but the domain of the original function is commonly referred to as the time domain. There is also an inverse Fourier transform that mathematically synthesizes the original function from its frequency domain representation, as proven by the Fourier inversion theorem.
Linear operations performed in one domain (time or frequency) have corresponding operations in the other domain, which are sometimes easier to perform. The operation of differentiation in the time domain corresponds to multiplication by the frequency, so some differential equations are easier to analyze in the frequency domain. Also, convolution in the time domain corresponds to ordinary multiplication in the frequency domain (see Convolution theorem). After performing the desired operations, transformation of the result can be made back to the time domain. Harmonic analysis is the systematic study of the relationship between the frequency and time domains, including the kinds of functions or operations that are "simpler" in one or the other, and has deep connections to many areas of modern mathematics.
Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution (e.g., diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation.
The Fourier transform can be formally defined as an improper Riemann integral, making it an integral transform, although this definition is not suitable for many applications requiring a more sophisticated integration theory. For example, many relatively simple applications use the Dirac delta function, which can be treated formally as if it were a function, but the justification requires a mathematically more sophisticated viewpoint. The Fourier transform can also be generalized to functions of several variables on Euclidean space, sending a function of 3-dimensional 'position space' to a function of 3-dimensional momentum (or a function of space and time to a function of 4-momentum). This idea makes the spatial Fourier transform very natural in the study of waves, as well as in quantum mechanics, where it is important to be able to represent wave solutions as functions of either position or momentum and sometimes both. In general, functions to which Fourier methods are applicable are complex-valued, and possibly vector-valued. Still further generalization is possible to functions on groups, which, besides the original Fourier transform on R or Rn (viewed as groups under addition), notably includes the discrete-time Fourier transform (DTFT, group = Z), the discrete Fourier transform (DFT, group = Z mod N) and the Fourier series or circular Fourier transform (group = S1, the unit circle ≈ closed finite interval with endpoints identified). The latter is routinely employed to handle periodic functions. The fast Fourier transform (FFT) is an algorithm for computing the DFT.
There is a Fourier transform that I don't really understand in my textbook.
I have the following equation:
##\ddot{\delta} + 2H\dot{\delta} -\frac{3}{2} \Omega_m H^2 \delta = 0##
Then using the Fourier transform:
##\delta_{\vec{k}} = \frac{1}{V} \int \delta(\vec{r}) e^{i \vec{k} \cdot \vec{r}}...
Suppose I have a pure sine wave. Upon Fourier transforming (FT) the time signal, I obtain a delta function in the frequency domain. If I subsequently modulate this sine wave with another function, for instance, a Gaussian, the delta function in the frequency domain will broaden. I'm curious...
Hello,
I understand how the FT and the STFT work. The STFT provides time-frequency localization, i.e. it can tell us when the spectral components are acting in the time-domain signal...The STFT is also useful for non-stationary signals which are signals whose statistical characteristics are...
I'm studying Quantum Field Theory for the Gifted Amateur and feel like my math background for it is a bit shaky. This was my attempt at a derivation of the above. I know it's not rigorous, but is it at least conceptually right? I'll only show it for bosons since it's pretty much the same for...
TL;DR Summary: Solve heat equation in a disc using fourier transforms
Carbon dioxide dissolves in the blood plasma but is not absorbed by red blood cells. As the blood returns to an alveolus, assume that it is well-mixed so that the concentration of dissolved CO2 is uniform across a...
So my first thought was that I can just use Fourier trick and integrate:
$$ F(q^2) = \int_V \rho(r) \cdot e^{ i \frac{ \vec{q} \cdot \vec{r} }{h} } d^3r $$
$$ F(q^2) = 2\pi \rho_0 \int_0^{\infty} r^2 \cdot e^\frac{-r}{R} dr \cdot \int_0^{\pi} \sin{\theta} \cdot e^{ -i \frac{q \cdot r...
My issue here is the fact that the slits are supposed to infinite in the ##y##-direction. With what's given in the assignment, I'd define the apparatus function ##a(x,y)## as
$$
a(x,y) = \begin{cases} 1 & , \, ( 9d \leq |x| \leq 10d ) \wedge (y \in \mathbb{R}) \\ 0 & , \, \text{else}...
Hey all,
I just wanted to double check my logic behind getting the Fourier Transform of the following Hamiltonian:
$$H(x) = \frac{ie\hbar}{mc}A(x)\cdot\nabla_{x}$$
where $$A(x) = \sqrt{\frac{2\pi\hbar c^2}{\omega L^3}}\left(a_{p}\epsilon_{p} e^{i(p\cdot x)} + a_{p}^{\dagger}\epsilon_{p}...
In my system I am trying to represent two lenses. L1 with focal length f1=910mm and the other lens, L2 with focal length f2=40mm. These lenses are space such that there is a distance of f1+f2 between the lenses. I have a unit amplitude plane wave incident on L1. My goal is to find the...
I was wondering if the following is true and if not, why?
$$
\begin{split}
\hat{f}_1(\vec{k}) \hat{f}_2(\vec{k}) &= \hat{f}_1(\vec{k}) \int_{\mathbb{R}^n} f_2(\vec{x}) e^{-2 \pi i \vec{k} \cdot \vec{x}} d\vec{x}
\\
&= \int_{\mathbb{R}^n} \hat{f}_1(\vec{k}) f_2(\vec{x})...
First,
##\tilde{f}(\omega)=\int_{-\infty}^{\infty}e^{a|t|}\cos(bt)e^{-i\omega t} \mathrm{d}t##
We can get rid of the absolute value by splitting the integral up
##\int_{-\infty}^{0}e^{at}\cos(bt)e^{-i\omega t} \mathrm{d}t+ \int_{0}^{\infty}e^{-at}\cos(bt)e^{-i\omega t} \mathrm{d}t##
Using...
I am unsure if ##h(x,t)## really is a wave packet, but it looks like one, hence the title. Anyway, so I'd like to determine ##\hat{h}(k,t=0)##. My attempt so far is recognizing that, without the real part in the integral, i.e.
##g(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty} a(k)e^{i(kx-\omega...
The 2D Fourier transform is given by: \hat{f}(k,l)=\int_{\mathbb{R}^{2}}f(x,y)e^{-ikx-ily}dxdy
In terms of polar co-ordinates: \hat{f}(\rho,\phi)=\int_{0}^{\infty}\int_{-\pi}^{\pi}rf(r,\theta)e^{-ir\rho\cos(\theta-\phi)}drd\theta
For Fourier transforms in cartesian co-ordinates, relating the...
I am unsure about what is being asked for in the question. At first I thought the question asks one to calculate the inverse Fourier transform and then to analyze its depends on ##t##, however, the "estimate" makes me think otherwise.
So,
##\hat{p}(\omega)=\int_{-\infty}^{\infty} p(t)e^{-i\omega t}\mathrm{d}t=A\int_{0}^{\infty}e^{-t(\gamma+i(\omega+\omega_0))}=A\left[-\frac{e^{-t(\gamma+i(\omega+\omega_0))}}{\gamma+i(\omega+\omega_0)}\right]_0^\infty,##
provided ##\gamma+ i(\omega+\omega_0)\neq 0## for the last equality. Now...
Greentings,
I've dealt with Quantum Theory a lot, but there's one thing I don't quite understand.
When deriving the Fermion-Propagator, say ##S_F##, all the authors I've read from made use of the Fourier-Transform. Basically, it always goes like
$$
\begin{align}
H_D S_F(x-y) &= (i \hbar...
I was studying a Michelson interferometer for infrared absorption in Fourier transform and I've found these two images (taken from https://pages.mtu.edu/~scarn/teaching/GE4250/ftir_lecture_slides.pdf ) representing an infrared monochromatic beam of light going into the interferometer and the...
Hello! I am reading about Fourier Transform MW spectroscopy in a FB cavity, which seems to be quite an old technique and I want to make sure I got it right.
As far as I understand, this is very similar to normal NRM, i.e. one applies a MW ##\pi/2## pulse which puts the molecules in a linear...
What is the Fourier transform of a beat? For example, I want to calculate the Fourier transform of the function ##f(t)=\cos((\omega_p+\omega_v) t)+\cos((\omega_p-\omega_v)t),## where ##$\omega_p+\omega_v=\Omega,\space\omega_p-\omega_v=\omega## and ##\Omega\simeq\omega.##
I think it is equal to...
Hello,
First of all, I checked several other threads mentioning duality, but could not find a satisfying answer, and I don't want to revive years old posts on the subject; if this is bad practice, please notify me (my apologies if that is the case).
For the past few days, I have had a lot of...
Doing the Fourier transform for the function above I'm getting a result, but since I can't get the function f(t) with the inverse Fourier transform, I'm wondering where I made a mistake.
##F(w) = \frac{1}{\sqrt{2 \pi}} \int_{0}^{\infty} te^{-t(a + iw)} dt##
By integrating by part, where G = -a...
My Professor has started on the Fourier Transforms Topic in the Introductory Mathematical Physics class and gave us a small homework to try our concepts on.
I have attached a clear & legible snippet of my solution. I request someone to please have a look at it & determine if my solution is...
Hello everybody!
I have a question concerning the Fourier transformation: So far I have experimentially measured the magnetic field of a quadrupole but as the hall effect sensor had a fixed orientation I did two series, one for the x, one for y component of the magnetic field, I have 50 values...
I have tried to Fourier transform in ##x## and get the result in the transformed coordinates, please check my result:
$$
\tilde{u}(k, y) = \frac{1-e^{-ik}}{ik}e^{-ky}
$$
However, I'm having some problems with the inverse transform:
$$
\frac{1}{2\pi}\int_{-\infty}^\infty...
Homework Statement:: .
Relevant Equations:: .
I am having a hard time thinking about Fourier transform, because there are so many conventions that i think i got more confused each time i think about it.
See an example, "Find the Fourier transform of $$V(t) = Ve^{iwt} \text{ if } nT \leq t...
Hello again. Having some issues on Fourier transform. Can someone please tell me how to proceed? Need to solve this then use some software to check my answer but how to solve for a and b. Plzz help
I am trying to understand the Helmholtz equation, where the Helmholtz equation can be considered as the time-independent form of the wave equation. It seems to me that the Helmholtz equation can be derived from the Fourier transform, such that it is part of a larger set of equations of varying...
I am trying to understand the relationship between Fourier conjugates in the spherical basis. Thus for two functions ##f(\vec{x}_3)## and ##\hat{f}(\vec{k}_3)##, where
\begin{equation}
\begin{split}
\hat{f}(\vec{k}_3) &= \int_{\mathbb{R}^3} e^{-2 \pi i \vec{k}_3 \cdot \vec{x}_3} f(\vec{x}_3...
I want to know the frequency domain spectrum of an exponential which is modulated with a sine function that is changing with time.
The time-domain form is,
s(t) = e^{j \frac{4\pi}{\lambda} \mu \frac{\sin(\Omega t)}{\Omega}}
Here, \mu , \Omega and \lambda are constants.
A quick...
The reference definition and problem statement are shown below with my work shown following right after. I would like to know if I am approaching this correctly, and if not, could guidance be provided? Not very sure. I'm not proficient at formatting equations, so I'm providing snippets, my...
I just want to make sure I am on the right track here (hence have not given the other information in the question). In taking the Fourier transform of the PDE above, I get:
F{uxx} = iω^2*F{u},
F{uxt} = d/dt F{ux} = iω d/dt F{u}
F{utt} = d^2/dt^2 F{u}
Together the transformed PDE gives a second...
I cite an original report of a colleague :
1) I can't manage to proove that the statistical error is formulated like :
##\dfrac{\sigma (P (k))}{P(k)} = \sqrt{\dfrac {2}{N_{k} -1}}_{\text{with}} N_{k} \approx 4\pi \left(\dfrac{k}{dk}\right)^{2}##
and why it is considered like a relative error ...
Would someone be able to explain like I am five years old, what is the precise relationship between Fourier series and Fourier transform?
Could someone maybe offer a concrete example that clearly illustrates the relationship between the two?
I found an old thread that discusses this, but I...
The definition of Fourier transform (F.T.) that I am using is given as:
$$f(\vec{x},t)=\frac{1}{\sqrt{2\pi}}\int e^{-i\omega t}\tilde{f}(\vec{x},\omega)\,\mathrm{d}\omega$$
I want to show that:
$$\frac{1}{c\sqrt{2\pi}}\int e^{-i\omega t}\omega^2...
Given ##f(\vec{x})##, where the Fourier transform ##\mathcal{F}(f(\vec{x}))= \hat{f}(\vec{k})##.
Given ##\vec{x}=[x_1,x_2,x_3]## and ##\vec{k}=[k_1,k_2,k_3]##, is the following true?
\begin{equation}
\begin{split}
\mathcal{F}(f(x_1))&= \hat{f}(k_1)
\\
\mathcal{F}(f(x_2))&= \hat{f}(k_2)
\\...
Hi
I would like to transform the S-parameter responce, collected from a Vector Network Analyzer (VNA), in time domain by using the Inverse Fast Fourier Transform (IFFT) . I use MATLAB IFFT function to do this and the response looks correct, the problem is that I do not manage to the time scaling...
I have not studied the Fourier transform (FT) in great detail, but came across a problem in electrodynamics in which I assume it is needed. The problem goes as follows:
Evaluate ##\chi (t)## for the model function...
In August, "Quantum Information Processing" published an article describing a full FFT in the quantum domain - a so-called QFFT, not to be confused with the simpler QFT.
According to the publication:
I am attempting to be able to do this problem with the help of Mathematica and Fourier transform. My professor gave us instructions for Fourier Transformation and Inverse Fourier, but I don't believe that my output in Mathematica is correct.
I was content with the understanding of the Fourier transform (FT) as a change of basis, from the time to the frequency basis or vice versa, an approach that I have often seen reflected in texts.
It makes sense, since it is the usual trick so often done in Physics: you have a problem that is...
Hi,
I was hoping to gain more insight into these window questions when looking at frequency spectra questions. I don't really know what makes windows better than one another.
My attempt:
In the question, we have f(t) = cos(\omega_0 t) and therefore its F.T is F(\omega ) = \pi \left(...
I have a question related to linearity of power spectral density calculation.
Suppose I have a time series, divided into some epochs. If I compute PSD by Welch's method with a time window equal to the length of an epoch and without any overlap, I obtain this result:
If I calculate the...
From the sketch, I know that this function is an even function. So, I simplify the Fourier transform in the limit of the integration (but still in exponential form). Then, I try to find the exponential FOurier transform. Here what I get:
$$g(a)=\frac{2}{2\pi} \int_{0}^{\infty} e^{-x} e^{-iax}...
https://en.wikipedia.org/wiki/Riemann_hypothesis#Quasicrystals a quasicrystal as "a distribution with discrete support whose Fourier transform also has discrete support."
https://en.wikipedia.org/wiki/Quasicrystal#Mathematicsdefines a quasicrystal as "a structure that is ordered but not...
Summary:: If ##f(x)=-f(x+L/2)##, where L is the period of the periodic function ##f(x)##, then the coefficient of the even term of its Fourier series is zero. Hint: we can use the shifting property of the Fourier transform.
So here's my attempt to this problem so far...
Hi,
In class I have learned how to find the Fourier transform of rectangular pulses. However, how do I solve a problem when I should sketch the Fourier transform of a pulse that isn't exactly rectangular.
For instance "Sketch the Fourier transform of the following 2 pulses"
Thanks in advance...
Given the domain of the integral for the Fourier transform is over the real numbers, how does the Fourier transform transform functions whose independent variable is complex?
For example, given
\begin{equation}
\begin{split}
\hat{f}(k_{\mathbb{C}}) &= \int_{\mathbb{R}} f(z_{\mathbb{C}})...